I liked the way they talked about science from a warrior race in Mass Effect. The Krogans have scientists but they’re mostly focused on making bigger bangs and booms. I would probably assume the Klingons are similar.
I know we have a few episodes showing other sides of the castes but generally we only interact with the warrior caste and occasionally see arbiters like in Rules of Engagement (DS9) or Judgement (ENT).
The only time we saw an actual Klingon scientist was in a TNG episode where they had figured out meta-phasic shields or something that allowed shuttles to get closer to a sun than ever before. Crusher was even dealing with some prejudice regarding Klingon scientists and has a few lines about how it feels weird to be working with a Klingon not focused on war and battle.